Question

AE3.8 Suppose that H1(s) and H2(s) are two strictly proper singleinput, single-output transfer functions with controllable statespace realizations (A1, B1, C1) and (A2, B2, C2), respectively. Construct a state-space realization for the parallel interconnection H1(s) + H2(s), and use the Popov-Belevitch-Hautus eigenvector test to show that the parallel realization is controllable if and only if A1 and A2 have no common eigenvalues.

Answer 1

We know that in a parallel interconnection the input is same for both blocks while the output gets added. We have used that fact to build a state equation for the composite parallel interconnection.The state matrix (A) and input matrix (B) of the parallel interconnection was used in the HPV test to prove the statement that A1 and A2 must have no common eigenvalues for the parallel interconnection to be controllable.